ID: math/9906167

Monstrous moonshine and the classification of CFT

June 25, 1999

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CFT, BCFT, ADE and all that

June 20, 2000

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Jean-Bernard Zuber
High Energy Physics - Theory

These pedagogical lectures present some material, classical or more recent, on (Rational) Conformal Field Theories and their general setting ``in the bulk'' or in the presence of a boundary. Two well posed problems are the classification of modular invariant partition functions and the determination of boundary conditions consistent with conformal invariance. It is shown why the two problems are intimately connected and how graphs -ADE Dynkin diagrams and their generalization...

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On classification of conformal vectors in vertex operator algebra and the vertex algebra automorphism group

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Yuto Moriwaki
Quantum Algebra

Herein we study conformal vectors of a Z-graded vertex algebra of (strong) CFT type. We prove that the full vertex algebra automorphism group transitively acts on the set of the conformal vectors of strong CFT type if the vertex algebra is simple. The statement is equivalent to the uniqueness of self-dual vertex operator algebra structures of a simple vertex algebra. As an application, we show that the full vertex algebra automorphism group of a simple vertex operator algebra...

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Algebraic Structures in Euclidean and Minkowskian Two-Dimensional Conformal Field Theory

February 22, 2009

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Liang Kong, Ingo Runkel
Mathematical Physics
Operator Algebras
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We review how modular categories, and commutative and non-commutative Frobenius algebras arise in rational conformal field theory. For Euclidean CFT we use an approach based on sewing of surfaces, and in the Minkowskian case we describe CFT by a net of operator algebras.

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Seventy Relatives of the Monster Module

April 21, 1993

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A. N. Schellekens
Quantum Algebra

Recent work on the classification of conformal field theories with one primary field (the identity operator) is reviewed. The classification of such theories is an essential step in the program of classification of all rational conformal field theories, but appears impossible in general. The last managable case, central charge 24, is considered here. We found a total of 71 such theories (which have not all been constructed yet) including the monster module. The complete list ...

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Conformal field theory and a new geometry

July 19, 2011

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Liang Kong
Quantum Algebra

This paper is a review of open-closed rational conformal field theory (CFT) via the theory of vertex operator algebras (VOAs), together with a proposal of a new geometry based on CFTs and D-branes. We will start with an outline of the idea of the new geometry, followed by some philosophical background behind this vision. Then we will review a working definition of CFT slightly modified from Segal's original definition and explain how VOA emerges from it naturally. Next, using...

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Classification of RCFT from Holomorphic Modular Bootstrap: A Status Report

October 7, 2019

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Sunil IISER Pune Mukhi
Number Theory

Following the initial proposal in 1988, there has been much progress in classifying Rational Conformal Field Theories in 2 dimensions from the Holomorphic Bootstrap approach. This method starts by postulating a generic holomorphic Modular Linear Differential Equation of a given order and imposing the requirement of non-negative integrality of the coefficients in the series expansion of the solutions, which are then identified as admissible characters, from which a modular-inv...

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Extension of Monster Moonshine to c=24k Conformal Field Theories

October 20, 2005

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Marcin Jankiewicz, Thomas W. Kephart
Mathematical Physics

We present a family of conformal field theories (or candidates for CFTs) that is build on extremal partition functions. Spectra of these theories can be decomposed into the irreducible representations of the Fischer-Griess Monster sporadic group. Interesting periodicities in the coefficients of extremal partition functions are observed and interpreted as a possible extension of Monster moonshine to c=24k holomorphic field theories.

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Rational Generalised Moonshine from Abelian Orbifoldings of the Moonshine Module

June 5, 2001

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Rossen I. Ivanov, Michael P. Tuite
Quantum Algebra
Group Theory

We consider orbifoldings of the Moonshine Module with respect to the abelian group generated by a pair of commuting Monster group elements with one of prime order $p=2,3,5,7$ and the other of order $pk$ for $k=1$ or $k$ prime. We show that constraints arising from meromorphic orbifold conformal field theory allow us to demonstrate that each orbifold partition function with rational coefficients is either constant or is a hauptmodul for an explicitly found modular fixing group...

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Conformal Field Theory, Tensor Categories and Operator Algebras

March 19, 2015

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Yasuyuki Kawahigashi
Mathematical Physics
Operator Algebras
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This is a set of lecture notes on the operator algebraic approach to 2-dimensional conformal field theory. Representation theoretic aspects and connections to vertex operator algebras are emphasized. No knowledge on operator algebras or quantum field theory is assumed.

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Topological modularity of Monstrous Moonshine

July 28, 2022

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Ying-Hsuan Lin
Algebraic Topology
Mathematical Physics
Number Theory
Representation Theory

We explore connections among Monstrous Moonshine, orbifolds, the Kitaev chain and topological modular forms. Symmetric orbifolds of the Monster CFT, together with further orbifolds by subgroups of Monster, are studied and found to satisfy the divisibility property, which was recently used to rule out extremal holomorphic conformal field theories. For orbifolds by cyclic subgroups of Monster, we arrive at divisibility properties involving the full McKay-Thompson series. Orbifo...

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